求导函数：
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
本次共计算 1 个题目：每一题对 z 求 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数{({R}^{2} + {(z + (\frac{a}{2}))}^{2})}^{\frac{-3}{2}} 关于 z 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{1}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{3}{2}}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{1}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{3}{2}}}\right)}{dz}\\=&(\frac{\frac{-3}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}})\\=&\frac{-3z}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}} - \frac{3a}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-3z}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}} - \frac{3a}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}}\right)}{dz}\\=&-3(\frac{\frac{-5}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}})z - \frac{3}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}} - \frac{3(\frac{\frac{-5}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}})a}{2} + 0\\=&\frac{15z^{2}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} + \frac{15az}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} + \frac{15a^{2}}{4(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{3}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{15z^{2}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} + \frac{15az}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} + \frac{15a^{2}}{4(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{3}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{5}{2}}}\right)}{dz}\\=&15(\frac{\frac{-7}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}})z^{2} + \frac{15*2z}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} + 15(\frac{\frac{-7}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}})az + \frac{15a}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} + \frac{15(\frac{\frac{-7}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}})a^{2}}{4} + 0 - 3(\frac{\frac{-5}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}})\\=&\frac{-105z^{3}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} - \frac{315az^{2}}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{45z}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{315a^{2}z}{4(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{45a}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{105a^{3}}{8(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-105z^{3}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} - \frac{315az^{2}}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{45z}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{315a^{2}z}{4(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{45a}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{105a^{3}}{8(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}}\right)}{dz}\\=&-105(\frac{\frac{-9}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}})z^{3} - \frac{105*3z^{2}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} - \frac{315(\frac{\frac{-9}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}})az^{2}}{2} - \frac{315a*2z}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + 45(\frac{\frac{-7}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}})z + \frac{45}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}} - \frac{315(\frac{\frac{-9}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}})a^{2}z}{4} - \frac{315a^{2}}{4(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{45(\frac{\frac{-7}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}})a}{2} + 0 - \frac{105(\frac{\frac{-9}{2}(0 + 2z + a + 0)}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}})a^{3}}{8} + 0\\=&\frac{945z^{4}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}} + \frac{1890az^{3}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}} - \frac{630z^{2}}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{2835a^{2}z^{2}}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}} - \frac{630az}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{945a^{3}z}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}} - \frac{315a^{2}}{2(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{9}{2}}} + \frac{945a^{4}}{16(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{11}{2}}} + \frac{45}{(R^{2} + z^{2} + az + \frac{1}{4}a^{2})^{\frac{7}{2}}}\\ \end{split}\end{equation} \]
你的问题在这里没有得到解决？请到 热门难题 里面看看吧！
新增加学习笔记(安卓版)百度网盘快速下载应用程序，欢迎使用。
新增加学习笔记(安卓版)本站下载应用程序，欢迎使用。
新增线性代数行列式的计算，欢迎使用。
数学计算和一元方程已经支持正割函数和余割函数，欢迎使用。
新增加贷款计算器模块（具体位置：数学运算 > 贷款计算器），欢迎使用。